Effective Dynamics of Local Observables for Extended Fermi Gases in the High-Density Regime

TL;DR

This paper rigorously derives the Hartree equation for high-density pseudo-relativistic Fermi gases, demonstrating local approximation of many-body dynamics and convergence of local observables in large domains at zero temperature.

Contribution

It extends previous results by proving local convergence of many-body dynamics to the Hartree equation for fermionic systems at high density and large domains.

Findings

Proves local convergence of many-body dynamics to Hartree dynamics.

Establishes bounds for local observables independent of density.

Uses local semiclassical bounds to control excitations.

Abstract

We give a rigorous derivation of the Hartree equation for the many-body dynamics of pseudo-relativistic Fermi systems at high density $\varrho \gg 1$, on arbitrarily large domains, at zero temperature. With respect to previous works, we show that the many-body evolution can be approximated by the Hartree dynamics locally, proving convergence of the expectation of observables that are supported in regions with fixed volume, independent of $\varrho$. The result applies to initial data describing fermionic systems at equilibrium confined in arbitrarily large domains, under the assumption that a suitable local Weyl-type estimate holds true. The proof relies on the approximation of the initial data through positive temperature quasi-free states, that satisfy strong local semiclassical bounds, which play a key role in controlling the growth of the local excitations of the quasi-free state along the many-body dynamics.